Exact sequences on Powell–Sabin splits


We construct smooth finite elements spaces on Powell–Sabin triangulations that form an exact sequence. The first space of the sequence coincides with the classical \(C^1\) Powell–Sabin space, while the others form stable and divergence-free yielding pairs for the Stokes problem. We develop degrees of freedom for these spaces that induce projections that commute with the differential operators.

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  1. 1.

    Alfeld, P.: A trivariate Clough–Tocher scheme for tetrahedral data. Comput. Aided Geomet. Des. 1(2), 169–181 (1984)

    Article  Google Scholar 

  2. 2.

    Arnold, D.N., Qin, J.: Quadratic velocity/linear pressure Stokes elements. In: Vichnevetsky, R., Knight, D., Richter, G. (eds.) Advances in Computer Methods for Partial Differential Equations–VII, pp. 28–34. IMACS (1992)

  3. 3.

    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 1–155, (2006)

  4. 4.

    Arnold, D .N., Falk, R .S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. (N.S.) 47(2), 281–354 (2010)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Christiansen, S., Hu, K.: Generalized finite element systems for smooth differential forms and Stokes’ problem. Numer. Math. (2018). https://doi.org/10.1007/s00211-018-0970-6

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Costabel, M., McIntosh, A.: On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. Math. Z. 265(2), 297–320 (2010)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Fu, G., Guzmán, J., Neilan, M.: Exact smooth piecewise polynomial sequences on Alfeld splits, arXiv:1807.05883 [math.NA], (2018)

  8. 8.

    Grošelj, J., Krajnc, M.: Marjeta, quartic splines on Powell–Sabin triangulations. Comput. Aided Geom. Des. 49, 1–16 (2016)

    Article  Google Scholar 

  9. 9.

    Grošelj, J., Krajnc, M.: Marjeta, \(C^1\) cubic splines on Powell–Sabin triangulations. Appl. Math. Comput. 272(1), 114–126 (2016)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Guzmán, J., Neilan, M.: Inf-sup stable finite elements on barycentric refinements producing divergence-free approximations in arbitrary dimension. SIAM J. Numer. Anal. 56(5), 2826–2844 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lai, M.-J., Schumaker, L.L.: Spline functions on triangulations, Encyclopedia of Mathematics and its Applications, 110. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  13. 13.

    Nédélec, J.-C.: A new family of mixed finite elements in \(R^3\). Numer. Math. 50(1), 57–81 (1986)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Software 3(4), 316–325 (1977)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. Math. Model. Numer. Anal. 9, 11–43 (1985)

    MATH  Google Scholar 

  16. 16.

    Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comp. 74(250), 543–554 (2004)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Zhang, S.: On the P1 Powell–Sabin divergence-free finite element for the Stokes equations. J. Comput. Math. 26(3), 456–70 (2008)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Zhang, S.: Quadratic divergence-free finite elements on Powell-Sabin tetrahedral grids. Calcolo 48(3), 211–244 (2011)

    MathSciNet  Article  Google Scholar 

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J. Guzman and A. Lischke were supported by the NSF grant DMS-1913083. M. Neilan was supported by the NSF grant DMS-1719829.

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Proof of Lemma 7


Suppose \(z \in W_r(\{a,m,b\})\) is such that (4.1a)–(4.1c) are all zero. We will show that z must be identically zero on [ab]. Let \(\psi (x)\) be a degree r polynomial on the interval [0, 1] satisfying

$$\begin{aligned} \begin{aligned} \psi (0)&= 1, \quad \psi (1) = 0, \\ \int _0^1 \psi (x) p(x)&= 0 \quad \forall p \in \mathcal {P}_{r-2}([0,1]). \end{aligned} \end{aligned}$$

We note that these conditions uniquely determine \(\psi\). Since z is continuous at m and equal to zero at a and b, and in view of (4.1b)–(4.1c), it follows that z may be represented by

$$\begin{aligned} z(y)&= z(m){\left\{ \begin{array}{ll} \psi \left( \frac{y-m}{a-m}\right) &{} y \in [a,m], \\ \psi \left( \frac{m-y}{m-b}\right) &{} y \in [m,b]. \end{array}\right. } \end{aligned}$$

Since \(z'(y)\) is continuous at m, it must hold that

$$\begin{aligned} \frac{-1}{m-b} \psi '(0) = \frac{1}{a-m}\psi '(0). \end{aligned}$$

Furthermore, given the conditions (A.1) on \(\psi\), we can show that \(\psi '(0) \ne 0\). Suppose that \(\psi '(0) = 0\) in addition to (A.1). Then for any \(p \in \mathcal {P}_{r-1}([0,1])\) with \(p(0) = 0\),

$$\begin{aligned} \int _0^1 \psi '(x) p(x)&= -\int _0^1 \psi (x) p'(x) + \psi (1) p(1) - \psi (0) p(0) = -\int _0^1 \psi (x) p'(x) = 0 \end{aligned}$$

since \(p'(x) \in \mathcal {P}_{r-2}([0,1])\). But \(\psi '(x)\) is itself such a function p(x), so it follows that

$$\begin{aligned} \int _0^1 |\psi '(x)|^2 = 0. \end{aligned}$$

Then \(\psi '(x) = 0\), and \(\psi\) is constant on [0, 1]. This contradicts (A.1), so \(\psi '(0) \ne 0\). Furthermore, since \(1/(b-m) \ne 1/(a-m)\), it follows that \(z(m) = 0\). Therefore \(z = 0\) on [ab]. \(\square\)

Proof of Theorem 3


(1) Proof of (4.10a). Let \(p \in C^\infty (T)\) and \(\rho := \text {rot }\varPi _0^r p - \varpi _1^{r-1} \text {rot }p \in S_{r-1}^1(T^{\mathrm{ps}})\). We show that \(\rho\) vanishes on (4.5).


$$\begin{aligned} \rho (z_i)&= \text {rot }\varPi _0^r p(z_i) - \varpi _1^{r-1} \text {rot }p(z_i) = 0,\\ \mathrm{div}\,\rho (z_i)&= -\mathrm{div}\,\varphi _1^{r-1} \text {rot }p(z_i) = -\mathrm{div}\,\text {rot }p(z_i) = 0, \end{aligned}$$

by the definitions of \(\varPi _0^r\) and \(\varpi _1^{r-1}\) along with DOFs (4.2a) and (4.5a).

Next, if \(r=2\),

$$\begin{aligned} \int _{e_i} \rho \cdot n_i&= \int _{e_i} \big (\text {rot }\varPi _0^r p - \varpi _1^{r-1} \text {rot }p \big )\cdot n_i\\&= \int _{e_i} \big (\text {rot }\varPi _0^r p - \varPi _1^{r-1} \text {rot }p \big )\cdot n_i=0, \end{aligned}$$

using (4.5b), (4.3b) and (4.7b). Similar arguments show that, for \(r\ge 3\),

$$\begin{aligned} \rho (z_{3+i})\cdot n_i&= (\text {rot }\varPi _0^r p (z_{3+i}) - \varPi _1^{r-1} \text {rot }p(z_{3+i}))\cdot n_i =0,\\ \int _e \rho \cdot w&= \int _e (\text {rot }\varPi _0^r p - \varpi _1^{r-1} \text {rot }p)\cdot w = \int _e (\text {rot }\varPi _0^r p - \varPi _1^{r-1} \text {rot }p)\cdot w = 0, \end{aligned}$$


$$\begin{aligned} \int _T \rho \cdot \text {rot }w&= \int _T (\text {rot }\varPi _0^r p - \varPi _1^{r-1} \text {rot }p) \cdot w = 0. \end{aligned}$$

Next using (4.5c) gives

$$\begin{aligned} \mathrm{div}\,\rho (z_{3+i}) = -\mathrm{div}\,\varpi _1^{r-1} \text {rot }p (z_{3+i}) = -\mathrm{div}\,\text {rot }p (z_{3+i}) = 0, \end{aligned}$$

and (4.5e) yields

$$\begin{aligned} \int _e (\mathrm{div}\,\rho ) q = -\int _e (\mathrm{div}\,\varpi _1^{r-1} \text {rot }p) q = -\int _e (\mathrm{div}\,\text {rot }p)q = 0 \end{aligned}$$

for all \(q\in \mathcal {P}_{r-4}(e)\) and \(e\in {\mathcal {E}}^b(T^{\mathrm{ps}})\). The same arguments, but using (4.5g), gives

$$\begin{aligned} \int _T (\mathrm{div}\,\rho ) q=0\qquad \forall q\in \mathring{L}_{r-1}^2(T^{\mathrm{ps}}). \end{aligned}$$

Applying Lemma 12 shows that \(\rho \equiv 0\), and so (4.10a) holds.

(2) Proof of (4.10b). For some \(v \in [C^\infty (T)]^2\), we define \(\rho := {\mathop {\mathrm {div}\,}}\varpi _1^{r-1} v - \varpi _2^{r-2} {\mathop {\mathrm {div}\,}}v \in L_{r-2}^2(T^{\mathrm{ps}})\). Then we need only show that \(\rho\) is zero for all DOFs in (4.6). For the vertex DOFs, we have for each \(z_i\),

$$\begin{aligned} \rho (z_i)&= {\mathop {\mathrm {div}\,}}\varpi _1^{r-1} v(z_i) - \varpi _2^{r-2} {\mathop {\mathrm {div}\,}}v(z_i) = 0, \end{aligned}$$

by (4.5a) and (4.6a). Next, for each \(i = 1,2,3\),

$$\begin{aligned} \rho (z_{3+i})&= {\mathop {\mathrm {div}\,}}\varpi _1^{r-1} v(z_{3+i}) - \varpi _2^{r-2} {\mathop {\mathrm {div}\,}}v(z_{3+i}) = 0, \end{aligned}$$

where we have used (4.5a) and (4.6b). Similar arguments show that

$$\begin{aligned} \int _e \rho q =0\qquad \forall q\in \mathcal {P}_{r-4}(e),\ e\in {\mathcal {E}}^b(T^{\mathrm{ps}}), \end{aligned}$$

by (4.5e) and (4.6c), and that

$$\begin{aligned} \int _T \rho q = 0\qquad \forall q\in \mathring{L}_{r-2}^2(T^{\mathrm{ps}}) \end{aligned}$$

by (4.5g) and (4.6e). Using (4.6d) and (4.5b) if \(r = 2\) or (4.5d) if \(r > 2\),

$$\begin{aligned} \int _T \rho&= \int _T {\mathop {\mathrm {div}\,}}\varpi _1^{r-1} v -\varpi _2^{r-2} {\mathop {\mathrm {div}\,}}v = \int _T {\mathop {\mathrm {div}\,}}(\varpi _1^{r-1} v - v) = \int _{{\partial }T} (\varpi _1^{r-1} v - v )\cdot n = 0. \end{aligned}$$

Therefore, \(\rho \equiv 0\) on T by Lemma 13, and (4.10b) is proved. \(\square\)

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Guzmán, J., Lischke, A. & Neilan, M. Exact sequences on Powell–Sabin splits. Calcolo 57, 13 (2020). https://doi.org/10.1007/s10092-020-00361-x

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  • Finite elements
  • Exact sequences
  • Commuting diagrams
  • Powell–Sabin triangulations

Mathematics Subject Classification

  • 65N30